Reliable Meaningful Communication

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Reliable Meaningful Communication

Madhu SudanMicrosoft, Cambridge,

USA

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Reliable Communication?

Problem from the 1940s: Advent of digital age.

Communication media are always noisy But digital information less tolerant to noise!

09/24/2013 HLF: Reliable Meaningful Communication 2

Alice Bob

We are not

ready

We are now

ready

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Theory of Communication

[Shannon, 1948]

Model for noisy communication channels Architecture for reliable communication Analysis of some communication schemes Limits to any communication scheme


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Channel = Probabilistic Map from Input to Output Example: Binary Symmetric Channel (BSC(p))

Modelling Noisy Channels


0 0

1 1

1-ppp

1-p

Input OutputChannel

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Some limiting values

p=0 Channel is perfectly reliable. No need to do anything to get 100% utilization

(1 bit of information received/bit sent)

p=½ Channel output independent of sender’s signal. No way to get any information through.

(0 bits of information received/bit sent)


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Lessons from Repetition

Can repeat (retransmit) message bits many times E.g., 0100 Decoding: take majority

E.g., 0110 Utilization rate = 1/3 More we repeat, more reliable the

transmission. More information we have to transmit, less

reliable is the transmission. Tradeoff inherent in all schemes? What do other schemes look like?


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Shannon’s Architecture

Sender “Encodes” before transmitting Receiver “Decodes” after receiving Encoder/Decoder arbitrary functions.

Rate = Hope: Usually


Alice BobEncoder Decoder

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Shannon’s Analysis

Coding Theorem: For every , there exists Encoder/Decoder that

corrects fraction errors with high probability with Rate

Binary entropy function:

monotone So if ; Channel still has utility! Note on probability: Goes to as


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Limit theorems

Converse Coding Theorem: If Encoder/Decoder have Rate then decoder

output wrong with prob.

Entropy is right measure of loss due to error. Entropy = ?

Measures uncertainty of random variable. (In our case: Noise).


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An aside: Data Compression

Noisy encoding + Decoding Message + Error (Receiver knows both). Total length of transmission Message length So is error-length ?

Shannon’s Noiseless Coding Theorem: Information (modelled as random variable) can

be compressed to its entropy … with some restrictions

General version due to Huffman


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1948-2013?

[Hamming 1950]: Error-correcting codes More “constructive” look at encoding/decoding

functions. Many new codes/encoding functions:

Based on Algebra, Graph-Theory, Probability. Many novel algorithms:

Make encoding/decoding efficient. Result:

Most channels can be exploited. Even if error is not probabilistic. Profound influence on practice.


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Modern Challenges


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New Kind of Uncertainty

Uncertainty always has been a central problem: But usually focusses on uncertainty introduced by the

channel Rest of the talk: Uncertainty at the endpoints (Alice/Bob)

Modern complication: Alice+Bob communicating using computers Both know how to program. May end up changing encoder/decoder

(unintentionally/unilaterally). Alice: How should I “explain” to Bob? Bob: What did Alice mean to say?


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Example Problem

Archiving data Physical libraries have survived for 100s of

years. Digital books have survived for five years. Can we be sure they will survive for the next

five hundred?

Problem: Uncertainty of the future. What formats/systems will prevail? Why aren’t software systems ever constant?

Problem: When designing one system, it is uncertain

what the other’s design is (or will be in the future)!


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Challenge:

If Decoder does not know the Encoder, how should it try to guess what it meant?

Similar example: Learning to speak a foreign language

Humans do … (?) Can we understand how/why? Will we be restricted to talking to humans only? Can we learn to talk to “aliens”? Whales?

Claim: Questions can be formulated mathematically. Solutions still being explored.


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Modelling uncertainty

Classical Shannon Model

09/24/2013

A B Channel

B2

Ak

A3

A2

A1 B1

B3

Bj

Uncertain Communication Model

New Class of ProblemsNew challenges

Needs more attention!

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Language as compression

Why are dictionaries so redundant+ambiguous? Dictionary = map from words to meaning For many words, multiple meanings For every meaning, multiple words/phrases Why?

Explanation: “Context” Dictionary:

Encoder: Context Meaning Word Decoder: Context Word Meaning Tries to compress length of word Should works even if Context1 Context2

[Juba,Kalai,Khanna,S’11],[Haramaty,S’13]: Can design encoders/decoders that work with uncertain context.


1

2

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A challenging special case

Say Alice and Bob have rankings of N movies. Rankings = bijections = rank of i th player in Alice’s ranking.

Further suppose they know rankings are close.

Bob wants to know: Is How many bits does Alice need to send (non-

interactively). With shared randomness – Deterministically?


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Meaning of Meaning? Difference between meaning and words

Exemplified in Turing machine vs. universal encoding Algorithm vs. computer program

Can we learn to communicate former? Many universal TMs, programming languages

[Juba,S.’08], [Goldreich,Juba,S.’12]: Not generically … Must have a goal: what will we get from the bits? Must be able to sense progress towards goal. Can use sensing to detect errors in understanding, and

to learn correct meaning. [Leshno,S’13]:

Game theoretic interpretation


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Communication as Coordination Game [Leshno,S.’13]

Two players playing series of coordination games Coordination?

Two players simultaneously choose 0/1 actions. “Win” if both agree:

Alice’s payoff: not less if they agree Bob’s payoff: strictly higher if they agree.

How should Bob play? Doesn’t know what Alice will do. But can hope to

learn. Can he hope to eventually learn her behavior and

(after finite # of miscoordinations) always coordinate?

Theorem: Not Deterministically (under mild “general” assumptions) Yes with randomness (under mild restrictions)09/24/2013 HLF: Reliable Meaningful Communication 21

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Summary

Understanding how to communicate meaning is challenging: Randomness remains key resource! Much still to be explored. Needs to incorporate ideas from many facets

Information theory Computability/Complexity Game theory Learning, Evolution …

But Mathematics has no boundaries …


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Thank You!


Reliable Meaningful Communication

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